8 LEBESGUE'S CONVERGENCE THEOREMS 131 (d) Let /i, /2 be real- valued functions on X such that ju,*(/i), ft*(/2), f<-*(/i), are all finite, Show that (If #2 is an integrable function such that f2 ^ g2 and ju.*(/2) = ju,(#2), observe that we have h g2 ^/i for every integrable function h such that h <*fi +/2-) Deduce that, if /i is integrable, then and f**(/i +/2)« M*(sup(/i,/2))+ ft*(inf(/i,/2)) (use (a)). (e) Let / be a finite integrable function, and let g be a function such that ^*(g) is finite. Show that g is integrable if and only if //,(/) = /x*(#) + //*(/ #). (If #1 is an integrable function such that g*^g\ and ft*(#i) = ft(#i), remark that 3. (a) For each pair of subsets A, B of X, show that (b) For each subset A of X, show that fc*(A) is the least upper bound of measures of compact sets contained in A. (c) For each subset A of X with finite outer measure, show that there exist two in- tegrable sets AI, A2 such that AI <= A <= A2 and /x#( A) = ft(Ai) and jLt*(A) = jtx(A2). Show that jit*(A - AI) = /x*(A2 A) = 0 and that ja*(A AO = /x*(A2 - A) = jLt*(A) - /x»(A). (d) Let A be an integrable set. Show that if B c A then (e) Let A, B, be disjoint sets of finite outer measure. If C = A u B, show that ^(A) + jit»(B) ^ ^(C) ^ /i*(A) + /x*(B) ^ /**() g /x*(A) and that g ]Lt*(A) + ft*(B)- /**( (To prove the latter inequality, reduce to the case where /^*(A) == ju*(B) = 0 with the help of (c): if A2, B2 are integrable sets such that A c A2 , B <= B2 and /x*(A) = ft(A2), jit*(B) = jLt(B2), show that ft*(C)^ /x(A3 n B2),) 4. Let A be Lebesgue measure on the unit interval I = [0, 1]. (a) Let a be a real number such that 0 <^ a <1. Construct a nowhere-dense (12.16) compact set in I whose Lebesgue measure is a. (Adapt the method of construction of the Cantor set (Section 4.2, Problem 2), for which a = 0.) (b) Construct a sequence (An) of mutually disjoint, nowhere-dense compact sets, such that A(An) =2~n and such that every interval which is a connected component of I I) At contains a subset of A»+i of measure >0, If A = U AB, show that A is i£n n a meager set of measure 1, and consequently that I A, which is the complement of a meager set, is \~negligible.gs of X into R.pact spaces and TT : X -> Y a proper continuous mapping